Problem: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $n \neq 0$. $p = \dfrac{n^2 - n - 72}{-6n + 54} \div \dfrac{-6n - 48}{n + 9} $
Dividing by an expression is the same as multiplying by its inverse. $p = \dfrac{n^2 - n - 72}{-6n + 54} \times \dfrac{n + 9}{-6n - 48} $ First factor the quadratic. $p = \dfrac{(n + 8)(n - 9)}{-6n + 54} \times \dfrac{n + 9}{-6n - 48} $ Then factor out any other terms. $p = \dfrac{(n + 8)(n - 9)}{-6(n - 9)} \times \dfrac{n + 9}{-6(n + 8)} $ Then multiply the two numerators and multiply the two denominators. $p = \dfrac{ (n + 8)(n - 9) \times (n + 9) } { -6(n - 9) \times -6(n + 8) } $ $p = \dfrac{ (n + 8)(n - 9)(n + 9)}{ 36(n - 9)(n + 8)} $ Notice that $(n - 9)$ and $(n + 8)$ appear in both the numerator and denominator so we can cancel them. $p = \dfrac{ \cancel{(n + 8)}(n - 9)(n + 9)}{ 36(n - 9)\cancel{(n + 8)}} $ We are dividing by $n + 8$ , so $n + 8 \neq 0$ Therefore, $n \neq -8$ $p = \dfrac{ \cancel{(n + 8)}\cancel{(n - 9)}(n + 9)}{ 36\cancel{(n - 9)}\cancel{(n + 8)}} $ We are dividing by $n - 9$ , so $n - 9 \neq 0$ Therefore, $n \neq 9$ $p = \dfrac{n + 9}{36} ; \space n \neq -8 ; \space n \neq 9 $